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Jiabin Zuo

Publications and source records attributed to Jiabin Zuo.

10 recordsLinked to original sources
arXiv2026

The uniqueness and concentration behavior of solutions for a nonlinear fractional Schr\"odinger system

The paper is concerned with a nonlinear system of two coupled fractional Schr\"odinger equations with both attractive intraspecies and attractive interspecies interactions in $\mathbb{R}$. By analyzing an associated $L^2$-constrained minimization problem, the uniqueness of solutions to this system is proved via the implicit function theorem. Under a certain type of trapping potential, by establishing some delicate energy estimates, we present a detailed analysis on the concentration behavior of the solutions as the total strength of intraspecies and interspecies interactions tends to a critical value, where each component of the solutions blows up and concentrates at a flattest common minimum point of the associated trapping potentials. An optimal blow-up rate of solutions to the system is also given.

math.AP↗
arXiv2024

Existence and non-existence of normalized solutions for a nonlinear fractional Schr\"odinger system

This paper is concerned with a nonlinear fractional Sch\"ordinger system in $\mathbb{{R}}$ with intraspecies interactions $a_{i}>0 \ (i=1,2)$ and interspecies interactions $\beta \in\mathbb{{R}}$. We study this system by solving an associated constrained minimization problem (i.e., $L^2-$norm constrains). Under certain assumptions on the trapping potentials $V_i(x) \ (i=1,2),$ we derive some delicate estimates for the related energy functional and establish a criterion for the existence and non-existence of solutions, in which way several existence results are obtained.

math.AP↗
arXiv2024

Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior

This paper is concerned with the ground states of a coupled pseudo-relativistic Hartree system in $\mathbb{R} ^{3} $ with trapping potentials, where the intraspecies and the interspecies interaction are both attractive. By investigating an associated constraint minimization problem, the existence and non-existence of ground states are classified completely. Under certain conditions on the trapping potentials, we present a precise analysis on the concentration behavior of the minimizers as the coupling coefficient goes to a critical value, where the minimizers blow up and the maximum point sequence concentrates at a global minima of the associated trapping potentials. We also identify an optimal blowing up rate under polynomial potentials by establishing some delicate estimates of energy functionals.

math.AP↗
arXiv2023

Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

In this paper, we investigate the following fractional Sobolev critical nonlinear Schrödinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-Δ)^{s} u=μ_{1} u+|u|^{2^{*}_{s}-2}u+η_{1}|u|^{p-2}u+γα|u|^{α-2}u|v|^β ~ \text{in}~ \mathbb{R}^{N},\\ (-Δ)^{s} v=μ_{2} v+|v|^{2^{*}_{s}-2}v+η_{2}|v|^{q-2}v+γβ|u|^α|v|^{β-2}v ~~\text{in}~ \mathbb{R}^{N},\\ \|u\|^{2}_{L^{2}}=m_{1}^{2} ~\text{and}~ \|v\|^{2}_{L^{2}}=m_{2}^{2}, \end{array}\right. \end{equation*} where $(-Δ)^{s}$ is the fractional Laplacian, $N={3,4}$, $s\in(0,1)$, $μ_{1}, μ_{2}\in\mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2^{*}_{s}$ is the fractional Sobolev critical index, $η_{1}, η_{2}, γ, m_{1}, m_{2}>0$, $α>1, β>1$, $p, q, α+β\in(2+4s/N,2^{*}_{s}]$. Firstly, if $p, q, α+β<2^{*}_{s}$, we obtain the existence of positive normalized solution when $γ$ is big enough. Secondly, if $p=q=α+β=2^{*}_{s}$, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.

math.AP↗
arXiv2022

Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-Δ)^{s(\cdot)}u+b(-Δ)u&=λ|u|^{-γ(x)-1}u+\left(\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u) & +ηH(u-α)|u|^{r(x)-2}u,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{align*} where $a(-Δ)^{s(\cdot)}+b(-Δ)$ is a mixed operator with variable order $s(\cdot):\mathbb{R}^{2N}\rightarrow (0,1)$, $a, b\geq 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\leq0$, $H(t) = 1$ if $t>0),$ $Ω\subset\mathbb{R}^N$ is a bounded domain, $N\geq 2$, $λ>0$, $0<γ^{-}=\underset{x\in\barΩ}{\inf}\{γ(x)\}\leqγ(x)\leqγ^+=\underset{x\in\barΩ}{\sup}\{γ(x)\}<1$, $μ$ is a continuous variable parameter, and $F$ is the primitive function of a suitable $f$. The variable exponent $r(x)$ can be equal to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$ with $\bar{s}(x)=s(x,x)$ for some $x\in\barΩ,$ and $η$ is a positive parameter. We also show that as $α\rightarrow 0^+$, the corresponding solution converges to a solution for the above problem with $α=0$.

math.AP↗
arXiv2022

Normalized ground state solutions for the fractional Sobolev critical NLSE with an extra mass supercritical nonlinearity

This paper is concerned with existence of normalized ground state solutions for the mass supercritical fractional nonlinear Schr\"{o}dinger equation involving a critical growth in the fractional Sobolev sense. The compactness of Palais-Smale sequences is obtained by a special technique, which borrows from the ideas of Soave (J. Funct. Anal. 279 (6) (2020) 1086102020). This paper represents an extension of previously known results - in the local and the nonlocal cases.

math.AP↗
arXiv2022

On critical variable-order Kirchhoff type problems with variable singular exponent

We establish a continuous embedding $W^{s(\cdot),2}(Ω)\hookrightarrow L^{α(\cdot)}(Ω)$, where the variable exponent $α(x)$ can be close to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\barΩ$. Subsequently, this continuous embedding is used to prove the multiplicity of solutions for critical nonlocal degenerate Kirchhoff problems with a variable singular exponent. Moreover, we also obtain the uniform $L^{\infty}$-estimate of these infinite solutions by a bootstrap argument.

math.AP↗
arXiv2020

Existence and multiplicity of positive weak solutions for a new class of $(p; q)$-Laplacian systems

The paper is concerned with the existence of positive weak solutions for a new class of $\left( p,q\right) $-Laplacian elliptic systems in a bounded domain by means of the method of sub-super solutions. Particularly, we do not need any sign conditions for $γ\left( 0\right), g\left( 0\right), f\left( 0\right) $ and $h\left(0\right) $. Moreover, a multiplicity result is obtained when $γ\left(0\right)=g\left( 0\right)=f\left( 0\right)=h\left( 0\right)=0.$ Finally, we give some examples to verify our main results.

math.AP↗
arXiv2020

A study of the Prandtl Batchelor problem using variational method

In this paper, we investigate the existence of nontrivial weak solutions for the Prandtl-Batchelor type free boundary value elliptic problem driven by a power nonlinearity. The algebraic topology approach will be used to establish the existence of solutions of approximate problem, while variational techniques will be used to determine the existence of major problem solutions. In the process several classical results are improved.

math.AP↗